A163733 Number of n X 2 binary arrays with all 1's connected, all corners 1, and no 1 having more than two 1's adjacent.

1, 1, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 1

Author

R. H. Hardin, Aug 03 2009

Keywords

• nonn

Comments

Same recurrence for A163695.

Same recurrence for A163714.

Appears to coincide with diagonal sums of A072405. - Paul Barry, Aug 10 2009

From Gary W. Adamson, Sep 15 2016: (Start)

Let the sequence prefaced with a 1: (1, 1, 1, 2, 2, 4, 6, ...) equate to r(x). Then (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) = the Fibonacci sequence, (1, 1, 2, 3, 5, ...). Let M = the following production matrix:

1, 0, 0, 0, 0, ...

1, 0, 0, 0, 0, ...

1, 1, 0, 0, 0, ...

2, 1, 0, 0, 0, ...

2, 1, 1, 0, 0, ...

4, 2, 1, 0, 0, ...

6, 2, 1, 1, 0, ...

...

Limit of the matrix power M^k as k->infinity results in a single column vector equal to the Fibonacci sequence. (End)

Apparently a(n) = A128588(n-2) for n > 3. - Georg Fischer, Oct 14 2018

Examples

			All solutions for n=8:
   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1
   0 1   1 0   1 0   1 0   1 0   1 0   0 1   0 1   0 1   0 1
   0 1   1 0   1 0   1 0   1 1   1 0   0 1   0 1   1 1   0 1
   0 1   1 0   1 0   1 1   0 1   1 0   0 1   0 1   1 0   1 1
   0 1   1 0   1 1   0 1   0 1   1 0   0 1   1 1   1 0   1 0
   0 1   1 0   0 1   0 1   0 1   1 1   1 1   1 0   1 0   1 0
   0 1   1 0   0 1   0 1   0 1   0 1   1 0   1 0   1 0   1 0
   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1
------
   1 1   1 1   1 1   1 1   1 1   1 1
   0 1   0 1   0 1   1 0   1 0   1 0
   1 1   1 1   0 1   1 0   1 1   1 1
   1 0   1 0   1 1   1 1   0 1   0 1
   1 1   1 0   1 0   0 1   0 1   1 1
   0 1   1 1   1 1   1 1   1 1   1 0
   0 1   0 1   0 1   1 0   1 0   1 0
   1 1   1 1   1 1   1 1   1 1   1 1
		

Links

• R. H. Hardin, Table of n, a(n) for n=1..100

Crossrefs

Cf. A118658, A055389, A006355, A006355, A169985, A000045.

Cf. A128588. - Georg Fischer, Oct 14 2018

Programs

Mathematica

Join[{1, 1}, Table[2*Fibonacci[n], {n, 70}]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
Table[Round[GoldenRatio^(k-1)] - Round[GoldenRatio^(k-1)/Sqrt[5]], {k, 1, 70}] (* Federico Provvedi, Mar 26 2013 *)

Formula

Empirical: a(n) = a(n-1) + a(n-2) for n >= 5.

G.f.: (1-x^3)/(1-x-x^2) (conjecture). - Paul Barry, Aug 10 2009

a(n) = round(phi^(k-1)) - round(phi^(k-1)/sqrt(5)), phi = (1 + sqrt(5))/2 (conjecture). - Federico Provvedi, Mar 26 2013

G.f.: 1 + 2*x - x*Q(0), where Q(k) = 1 + x^2 - (2*k+1)*x + x*(2*k-1 - x)/Q(k+1); (conjecture), (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013

G.f.: If prefaced with a 1, (1, 1, 1, 2, 2, 4, ...): (1 - x^2 - x^4)/(1 - x - x^2); where the modified sequence satisfies A(x)/A(x^2), A(x) is the Fibonacci sequence. - Gary W. Adamson, Sep 15 2016